# A locally supersymmetric $SO(10,2)$ invariant action for $D=12$ supergravity

## TL;DR

This paper constructs a supersymmetric action for supergravity in 12 dimensions with $SO(10,2)$ invariance, using a gauge superalgebra that is only partially gauge-invariant, and includes the Einstein-Hilbert term.

## Contribution

It introduces a novel $D=12$ supergravity action with $SO(10,2)$ invariance based on a subalgebra of $OSp(1|64)$, incorporating off-shell supersymmetry.

## Key findings

- The action is invariant under a specific subalgebra ${ ilde F}$ of $OSp(1|64)$.
- Supersymmetry transformations close off-shell.
- The action contains the Einstein-Hilbert term.

## Abstract

We present an action for $N=1$ supergravity in $10+2$ dimensions, containing the gauge fields of the $OSp(1|64)$ superalgebra, i.e. one-forms $B^{(n)}$ with $n$=1,2,5,6,9,10 antisymmetric D=12 Lorentz indices and a Majorana gravitino $\psi$. The vielbein and spin connection correspond to $B^{(1)}$ and $B^{(2)}$ respectively. The action is not gauge invariant under the full $OSp(1|64)$ superalgebra, but only under a subalgebra ${\tilde F}$ (containing the $F$ algebra $OSp(1|32)$), whose gauge fields are $B^{(2)}$, $B^{(6)}$, $B^{(10)}$ and the Weyl projected Majorana gravitino ${1 \over 2} (1+\Gamma_{13}) \psi$. Supersymmetry transformations are therefore generated by a Majorana-Weyl supercharge and, being part of a gauge superalgebra, close off-shell. The action is simply $\int STr ({\bf R}^6 {\bf \Gamma})$ where ${\bf R}$ is the $OSp(1|64)$ curvature supermatrix two-form, and ${\bf \Gamma}$ is a constant supermatrix involving $\Gamma_{13}$ and breaking $OSp(1|64)$ to its ${\tilde F}$ subalgebra. The action includes the usual Einstein-Hilbert term.